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Let's say you have a sphere-maze.

enter image description here

The sphere is fixed on the fixed half-ring (1) and two independent rings (2) and (3). These rings are connected at point A. Inside the sphere one can see a small ball (point B). When you are rotate two rings, the ball begins to spin.

Suppose that the initial position of the sphere is the combination of a half ring (3), two rings (1, 2) and a red great circle (point G in the photo) on one plate.

On the sphere, the sticker is placed. The player must rotate the sphere so as to place the ball on the sticker. The angle between the rings (2) and (3) is a hint for next puzzle.

Question. How to check (prove) that the puzzle has an unique solution?

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A standard 3-axis gimbal set has a unique state (yaw, pitch, and roll) for each orientation in space except for two points where you get gimbal lock. Think of a fighter jet - yaw is the compass direction it is travelling, pitch is the upwards angle relative to the ground, and roll is the rotation around its own axis i.e. whether one wing is dipped below the other. Gimbal lock occurs when pitch is vertical, at which point yaw and roll coincide.

You seem to only need two of the coordinates (pitch, roll). Once the ball is on the spot, the rotation of the sphere about the vertical axis (yaw) does not matter. This rotation only affects the outer gimbal (#2 relative to frame #1). The angles between #2 and #3 (pitch), and between #3 and G (roll) are uniquely determined unless the sticker/ball are on the axis where G is connected to #3 where gimbal lock occurs (but even then only roll is indeterminate, the pitch angle between #2 and #3 will be 90 degrees).

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